On the existence theorem for weak solutions of increased smoothness of the multidimensional Navier-Stokes equations (Q1874764)

The author examines multidimensional flows of an incompressible viscous fluid with velocities periodic in each space variable. It is shown that, if the initial velocity is square integrable over the period cube, its derivatives are integrable, and the mass forces satisfy similar integrability conditions over the space-time parallelepiped, then there exist weak solutions whose second derivatives are \((4/3-\varepsilon)\)th-power integrable over the space-time parallelepiped for any \(\varepsilon> 0\). The proof is based on a priori estimates and a compactness argument.